Bar Chart Confirmed

Bar Chart Table Deaths

Table Confirmed and Deaths - Cumulated and Daily Cases

Bar Chart Confirmed per 100,000 Inhabitants

Bar Chart Deaths per 100,000 Inhabitants

Table Confirmed and Deaths - Cumulated and Daily Cases per 100,000 Inhabitants

World

Exponential Growth Evaluation

China and South Korea have slowed down significantly the exponential growth. Therefore, their lines in the chart with the log10 scale have no longer a significant slope.

Most other countries are still in a phase of more or less unchecked exponential growth. For Italy, the reduced exponential growth is reflected in a reduced slope of the cumulated cases.

Virus Spread with log10 scale (since mid of Jan)

Virus Spread / Inhabitants with log10 scale (since mid of Jan)

Estimation spread of the Coronavirus with Linear Regression of log data

Exponential Growth and Doubling Time \(T\)

Exponential growth over time can be fitted by linear regression if the logarithms of the case numbers is taken. Generally, exponential growth corresponds to linearly growth over time for the log (to any base) data.

The semi-logorithmic plot with base-10 log scale for the Y axis shows functions following an exponential law \(y(t) = y_0 * a^{t/\tau}\) as straight lines. The time constant \(\tau\) describes the time required for y to increase by one factor of \(a\).

If e.g. the confirmed or death cases are growing in \(t-days\) by a factor of \(10\) the doubling time \(T \widehat{=} \tau\) can be calculated with \(a \widehat{=} 2\) by

\(T[days] = \frac {t[days] * log_{10}(2)} {log_{10}(y(t))-log_{10}(y_0)}\)

with

\(log_{10}(y(t))-log_{10}(y_0) = = log_{10}(y(t))/y_0) = log_{10}(10*y_0/y_0) = 1\)

and doubling time

\(T[days] = t[days] * log_{10}(2) \approx t[days] * 0.30\).

For Spain, Italy, Germany we have had a doubling time up to \(T \approx 9-12 days * 0.3 \approx 2.7 - 4 days !!\).

The doubling time \(T\) and the Forecast is calculated for following selected countries: Austria, France, Germany, Italy, India, South Korea, Spain, United States of America

Germany - Trend with Forecast on a linear scale

Forecast Plot - next 14 days

The plot shows the extreme forecast increase in case of unchecked exponentiell growth. The dark shaded regions show 80% rsp. 95% prediction intervals. These prediction intervals are displaying the uncertainty in forecasts based on the linear regression over the past 7 days.

Comparison Exponential Growth

Germany - Example plot with ~linear slope on a log10 scale

Comparison Exponential vs. Linear Growth

The charts compare the different forecasts for an exponential rsp. linear growth model. Due to the large fluctuations of the daily cases regression of three weeks is required. Otherwise the prediction levels are much too big.

The dark shaded regions are indicating the \(80\%\) rsp. \(95\%\) prediction intervals. These prediction intervals are displaying the “pure” statistical uncertainty in forecasts based on the regression models.

For doubling periods in the order of infectivity (RKI assumption: \(\sim9-10\) days, with great uncertainty), https://www.rki.de/DE/Content/InfAZ/N/Neuartiges_Coronavirus/Modellierung_Deutschland.pdf?__blob=publicationFile), we no longer have exponential growth. The “old” infected cases are at the end of the doubling period no longer infectious (active). This results in a constant infection rate with basic reproduction number \(R_0 \sim 1\) or even \(<1\).

Cumulated Cases - Comparison Exponential and Linear Growth

Daily Cases - Comparison Exponential and Linear Growth

Check of Forecast Accuracy

The forecast accuracy is checked by using the forecast method for the nine days before the past three days (training data). Subsequent forecasting of the past three days enables comparison with the real data of these days (test data).

The comparison is also an early indicator if the exponential growth is declining. However, possible changes in underreporting (in particular the proportion confirmed / actually infected) requires careful interpretation.

For doubling periods in the order of infectivity (RKI assumption: \(\sim9-10\) days, with great uncertainty), https://www.rki.de/DE/Content/InfAZ/N/Neuartiges_Coronavirus/Modellierung_Deutschland.pdf?__blob=publicationFile), we no longer have exponential growth. Since the “old” infected cases are no longer infectious after these periods and we then have a constant infection rate with basic reproduction number \(R_0 \sim 1\).

Instead, we have “only” linear growth of the cumulative Confimred Cases and the Daily Confirmed Cases remain more or less constant or even decrease.

However, the basic reproduction number \(R_0\) is a product of the average number of contacts of an infectious person per day, the probability of transmission upon contacts and the average number of days infected people are infectious. With the current uncertainty of the average duration of the infectivity duration, \(R_0\) can therefore be estimated from the doubling time only to a very limited extent. See also: https://cmmid.github.io/topics/covid19/current-patterns-transmission/global-time-varying-transmission

Germany - Forecast Accuracy for past three days

Forecasting with lagged Predictors

The number of confirmed cases can be used as a time delayed predictor of the number of deaths. This will allow comclusions on the time period confirmed to death. More inportant the country specific case fatality rate (CFR, proportion of deaths from confirmed cases) indicates the country specific testing.

Overall a rough conclusion on the country specific proportion of infected to confirmed cases is feasible if the infection fatality rate (IFR, confiremd cases plus all asymptomatic and undiagnosed infections) is assumed to be country independent and the IFR is known (bottom of existing estimates \(\sim0.56\%\), assumption by RKI see https://www.rki.de/DE/Content/InfAZ/N/Neuartiges_Coronavirus/Modellierung_Deutschland.pdf?__blob=publicationFile ).

Therefore an estimation of the CFR of \(0.06\) \((6\%)\) indicates an underreporting or lack of diagnostic confirmation by a by a factor of \(\sim10\). A CFR of \(0.20\) \((20\%)\) indicates an underreporting by a by a factor of \(\sim30\). This corresponds to RKI assumption of a underreporting by a factor of \(11-20\) (https://www.rki.de/DE/Content/InfAZ/N/Neuartiges_Coronavirus/Steckbrief.html).

In the model paper RKI assumes for the

• Incubation period \(\sim5-6\) days - Day of infection day until symptoms are upcoming)
• Hospitalisation \(+4\) days - Admission to the hospital (if needed) after Incubation Period)
• Average period to death \(+11\) - if the patient dies, it takes an average of \(11\) days after admission to the hospital

Depending on the country-specific test frequency (late or early tests), the

*lag_days - time from receipt of the confirmed test result to death, Confirmed to Death, is about \(11-13\) days.

Note: these methods are also used for example for advertising campaigns. The campaign impact on sales will be some time beyond the end of the campaign, and sales in one month will depend on the advertising expenditure in each of the past few months (see https://otexts.com/fpp3/lagged-predictors.html).

Daily Confirmed and Death Cases

Lag days and Case Fatality Rate (CFR)

Lag days and CFR (proportion of deaths from confirmed cases)

Country	lag_Confirmed	CFR
Germany	12	0.04
Italy	12	0.06
Spain	10	0.03

Forecast residuals indicate quality of fit with Arima model:

ARIMA(Deaths ~ 0 + pdq(d = 0) + lag(Confirmed, lag_days - 2) + lag(Confirmed, lag_days - 1) + lag(Confirmed, lag_days) + lag(Confirmed, lag_days + 1) + lag(Confirmed, lag_days + 2)) .

Daily Deaths depending on lagged Daily Confirmed Cases

Exampla Germany - White Noise of Forecast Residuals